Projectile Motion Calculator

How Projectile Motion Works

Once a projectile leaves the launcher, only gravity acts on it (this calculator ignores air resistance). That means its horizontal velocity never changes — it just keeps moving sideways at a constant speed — while its vertical velocity is being constantly pulled downward by gravity, exactly like an object in free fall. Combine constant horizontal motion with accelerating vertical motion and you get a parabola.

The Key Formulas

$$x(t) = v_0 \cos(\theta) \, t \qquad y(t) = y_0 + v_0 \sin(\theta) \, t - \frac{1}{2} g t^2$$

Maximum height is reached when vertical velocity hits zero, at $t = \frac{v_0 \sin(\theta)}{g}$, giving a height of $y_0 + \frac{(v_0 \sin(\theta))^2}{2g}$. Flight time comes from solving $y(t) = 0$ for $t$ using the quadratic formula.

Why 45° Is the "Best" Angle — With a Catch

For a given launch speed, 45° maximizes range only when the projectile lands at the same height it launched from. Launch from an elevated platform (y₀ > 0) and the optimal angle drops slightly below 45°, since a flatter shot gets more of its speed working horizontally for longer before gravity finally brings it down.

Worked Example: Launched at 20 m/s and 30°

For a launch speed of 20 m/s at 30° from level ground ($y_0 = 0$), flight time is $t = \frac{2 v_0 \sin(\theta)}{g} = \frac{2 \times 20 \times 0.5}{9.8} \approx 2.04$ s, maximum height is $\frac{(20 \times 0.5)^2}{2 \times 9.8} \approx 5.10$ m, and — using the range shortcut for level ground, $R = \frac{v_0^2 \sin(2\theta)}{g}$ — the range is $\frac{400 \times \sin(60°)}{9.8} \approx 35.35$ m. That range formula only works when launch and landing heights are equal; from an elevated platform you need the full quadratic solve this calculator performs.

Reading the Animated Trajectory

Press Play to watch the shot fly in real time, or drag the scrubber to jump straight to any instant — the chart below stays in sync, showing exactly where the projectile is on its parabola. Changing any input recalculates the trajectory immediately; pressing Play always restarts the animation from launch using the current values.

A Brief History of Projectile Motion

Understanding projectile motion mattered enormously to early ballistics and artillery, and much of the field's early progress came from that practical need. Italian mathematician Niccolò Tartaglia published influential work on cannon trajectories in the 1530s and 1540s, arguing (correctly, though without full proof) that a 45-degree launch angle maximizes range — a conclusion Galileo Galilei later proved rigorously through his broader study of motion in the early 1600s. Galileo's key insight was that projectile motion could be decomposed into two independent components — constant horizontal velocity and uniformly accelerated vertical motion under gravity — which combine to trace a parabola. This decomposition, still the foundation of the equations used above, was a major conceptual breakthrough because it let two previously separate problems (horizontal travel and vertical fall) be solved independently and then combined.

Common Projectile Motion Mistakes

Forgetting that horizontal and vertical motion are independent is the most fundamental error — gravity only affects the vertical component, so horizontal velocity stays constant throughout the flight (ignoring air resistance) even as vertical velocity continuously changes. Assuming 45 degrees always gives the maximum range is another common oversimplification: that result only holds when launch and landing height are the same, and the optimal angle shifts below 45 degrees when launching from an elevated position, as covered in the FAQ below. Confusing speed and velocity — treating the vector direction as irrelevant — can also lead to errors when combining the horizontal and vertical components back into total impact speed.

Projectile Motion Terms You Should Know

Trajectory — the parabolic path a projectile follows under constant gravitational acceleration and no air resistance.

Range — the total horizontal distance a projectile travels before landing.

Apex — the highest point of the trajectory, where vertical velocity momentarily equals zero.

Time of Flight — the total duration the projectile is in the air, from launch to landing.